252 HlSTOKT OF THE THEORY OF NUMBERS. [CHAP, VIII the residues prime to p, <t>(x) constitute the direct product of the groups with respect to the various p, <t>i(x) . Not every abelian group can be represented as a congruence group composed of a complete set of prime residues with respect to FI, . . . , Fx, where the F's are functions of a single variable. Mildred Sanderson115 employed two moduli m and P(y), the first being any integer and the second any polynomial in y with integral coefficients. Such a polynomial f(y) is said to have an inverse fi (y) if ffi= 1 (mod m, P). If P(y) is of degree r and is irreducible with respect to each prime factor of w, a function f(y), whose degree is <r, has an inverse modulis m, P(y), if and only if the g. c. d. of the coefficients of f(y) is prime to m. For such an /, /n==l (mod m, P), where n is Jordan's function Jr(m) [Jordan,200 Ch. V]. In case m is a prime, this result becomes Galois'62 generalization of Fermat's theorem. The product of the n distinct residues having inverses modulis w, P(y), is congruent to —1 when m is a power of an odd prime or the double of such a power or when r = l, m = 4; but congruent to +1 in all other cases — a two-fold generalization of Wilson's theorem. There exists a polynomial P(y) of degree r which is irreducible with respect to each prime factor of m. Then if A(y), B(y) are of degrees <r and their coefficients are not all divisible by a factor of m, there exist polynomials a(t/), P(y), such that aA-H3JS=l (mod ra, P). Several writers116 discussed the irreducible quadratic factors modulo p of (xa — l)/(xh — 1), where k = l or 2, p is a prime, a a divisor of p+l. G. Tarry117 noted that, if j2z=q (mod m), where q is a quadratic non-residue of the prime m, the Galois imaginary a+bj is a primitive root if its norm (a +6?) (a — bf) is a primitive root of m and if the ratio a:b and the analogous ratios of the coordinates of the first m powers of a +bj are incon-gruent. L. E. Dickson118 proved that two polynomials in two variables with integral coefficients have a unique g. c. d. modulo p, a prime. Thus the unique factorization theorem holds. G. Tarry119 stated that Ap is a primitive root of the GF[p*] if the norm of A = a+bj is a primitive root of p and if the imaginary p belongs to the exponent p+1. The 0(p + l) numbers p are found by the usual process to obtain the primitive roots of a prime. U. Scarpis120 proved that an equation of degree v irreducible in the Galois field of order pn has in the field of order pmn either v roots or no root according as v is or is not a divisor of m [Dickson102, p. 19, lines 7-9]. CUBIC CONGRUENCES. A. Cauchy130 solved y*+ By-\-C^O (mod p) when it has three distinct "B Annals of Math., (2), 13, 1911, 36-9. "8L'interm6diaire dea math., 18, 1911, 195, 246; 19, 1912, 61-69, 95-6; 21, 1914, 158-161; 22, 1915, 77-8. Sphinx-Oedipe, 7, 1912, 2-3. »7Assoc. fran$. av. sc., 40, 1911, 12-24. 118Bull. Amer. Math. Soc., (2), 17, 1911, 293-4. ""Sphinx-Oedipe, 7, 1912, 43-4, 49-50. "°Annali di Mat., (3), 23, 1914, 45. 130Exercices de Math., 4, 1829, 279-292; Oeuvres, (2), 9, 326-333.